I 'm not sure of what you say.
Doing this with Monte Carlo random numbers (Box-Muller) does n't seem to help.
With n = 2 and 100,000 samples it even finds errors, P(2) > P(1) and we know that for n=2, P(2) < P(1).
The proof for n = 2 exists somewhere but I don't have the proof that the order 1...
There are a number n of runners in a race.
We know their expected times from start to finish μ(i) and the corresponding standard deviations σ(i).
The probability of runner 0 to finish first is given by this integral:
It's from here:
https://www.untruth.org/~josh/math/normal-min.pdf
The 0 is...
Now I rememeber I did something like that in 2008.
The European nations football cup tournament in Austria-Switzerland.
With 16 teams and pair to pair eliminations I got something. It only needed tenths of seconds to compute.
So powerful Spain was indeed the favourite and possible closeups were...
I did not reject anything.
I know the ELO ratings.
I just said that -somehow- we know the head to head probabilities.
The ELO ratings apply to football.
From those we derive probabilities (not great stuff in terms of achieving the most accurate of predictions but acceptable).
The anomaly...
I think I found what went wrong.
With the Q values (pair to pair) we find an approximation.
The Monte Carlo method / integral are more precise.
What goes on ?
Consider the following thought experiment:
The race between 1, 2 and 3 takes place and is recorded on video.
A tv assistant photoshops...
For the original problem I said (AB)C, (BC)A, (AC)B are equivalent arrangements.
This gives probabilities that add to 1, but it's a false indicator. Therein must lie the error.
So is there no way to solve the simple problem if we know the head to head probabilities but we know nothing about the...
Hmm.
I used Laplace for the only reason that it computes the integral analytically for two contestants (and maybe for more but it gets difficult).
This I did together with Monte Carlo two posts above to verify my Bayesian (or quasi-Bayesian) approach.
Strangely it does n't verify.
Have I made...
From some distribution he says.
Normal is the logical thing to assume, could also be flat topped.
What is a latent trait ?
But it baffles me now.
Forget the horses I mention before and say we throw javelins.
Me and my friends Tom and Dick.
We have all been training for the past year and our...
The way the simple problem is stated in the first post it's not readily amenable to simulation using Monte Carlo.
But I can do it using the integral in the link of post no 3.
Suppose it concerns three race horses running a race of 5 furlongs.
We know from past measurements their mean times from...
No.
Forget how these head to head probabilities are derived (elo, speed distributions, whatever).
We assume they are independent.
Perhaps not in real life, but for our purposes they are.
If you talk about races, in real life the jockey sees the rest of the field stretches and miles behind him...
Look, suppose it's a race.
The winning attribute is speed and it may follow a normal distribution if we go into the details.
So if we ignore psychological factors it's going to be the same with or without C present. So the rest follow.
Look at this...
Is there a formula for this ?
Consider the following simple looking problem.
We have three contestants A, B and C, there is a competition between them and the best wins.
For example a race, discus throw, javelin throw ...
We know that A beats B with a probability 0.6, A beats C with a...
It's complicated.
Michael Faraday claimed copyright for electricity but he lost his case. The judge said to him you cannot own a natural force.
But Pirelli won their case against Michelin in the sixties, because the courts opined Pirelli copied the French and the design of the car tire was so...
There is a law that says "the worst is yet to come". It's in wikipedia also.
So that means if the tallest man on Earth is 2.40 meters tall, if we wait long enough somebody 3 meters tall will turn up.
I have in fact tried this using random sample from the normal distribution and noted the extreme...